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I came across a question in which speed distribution of molecules of an ideal gas trapped in a vessel and the temperature of the gas was given. Then it was given that the molecules of a particular speed escape from the vessel and we had to determine the new temperature of the gas. I just by trial and error figured out that the new temperature could be obtained if I assumed that the root mean square speed of the gas divided by the root of temperature that is $$v_{rms}/(T)^{0.5}$$ remained constant which by formula is equal to $(3R/M)^{0.5}$ and it makes sense.

But I don’t understand that why is it only the root mean square speed of the gas that remains constant why does say average velocity or the most probable speed not remain constant?

For a clearer idea of what I am asking, I am also attaching the question : Speed distribution of molecules of an ideal mono atomic gas trapped in a vessel is given in the following table.

Speed of molecules Percentage of molecules

100 m/s                     10
200 m/s                     20
600 m/s                     40
800 m/s                     20
1000 m/s                    10

Temperature of this gas is 324 K. If all the molecules of speed 800 m/s escape from the vessel, by what amount will the temperature of the gas change?

Any help would be greatly appreciated!

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Because the heat energy remains the same and there is no expansion work done (expansion of gas while applying a force on a surface via pressure). This means that total energy is conserved. The energy of a molecule is its kinetic energy, 1/2mv^2, and the total energy is the sum of the kinetic energy of every molecule. To keep this sum from changing, the root mean square must be constant.

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